When the same sound is produced simultaneously with two different fundamental frequencies, auditory roughness is observed. If the first sound is fixed and the fundamental frequency of the second is varied continuously, auditory roughness also varies continuously. A vowel sound is distinguished by its spectral envelope – which is independent of the fundamental frequency. This is a motivation to define the metric space of timbres. Each timbre is associated with a dissonance function which has local minima at certain intervals of local consonance related to the timbre. This is related to the music-theoretical notion of consonant intervals and scales. For the subspace consisting of all timbres with an interval of local consonance at a chosen point β, the main theorem describes certain points on the boundary by the vanishing of one-sided derivatives of dissonance functions at β.

当用两个不同的基频同时产生相同的声音时，会观察到听觉粗糙度。如果第一个声音是固定的，而第二个声音的基频连续变化，则听觉粗糙度也会连续变化。元音的特征在于其频谱包络——它与基频无关。这是定义音色度量空间的动机。每个音色都与一个不协和函数相关联，该函数在与音色相关的局部协和的特定间隔处具有局部最小值。这与辅音音程和音阶的音乐理论概念有关。对于由所有音色组成的子空间，在选定点β处具有局部协和音程, 主要定理通过β处失调函数的单边导数消失来描述边界上的某些点。